Solving Logarithmic Equations: Log Base 3 (x+3) = 2

Hey guys! Let's dive into solving a logarithmic equation today. We're going to break down the problem step-by-step so you can easily understand how to tackle these types of questions. Our specific problem is: log base 3 of (x+3) = 2. Let's get started!

Understanding Logarithmic Equations

Before we jump into solving, it’s super important to understand what a logarithm actually is. A logarithm is essentially the inverse operation to exponentiation. Think of it this way: the logarithm answers the question, “To what power must we raise the base to get this number?” In our case, we have log base 3 of (x+3) = 2. This means we're asking, “To what power must we raise 3 to get (x+3)?” Grasping this concept is the first step in confidently solving any logarithmic equation. So, make sure you're solid on the basics before moving forward!

Converting Logarithmic to Exponential Form

The key to solving most logarithmic equations is to convert them into their equivalent exponential form. Remember, guys, a logarithm is just another way of expressing an exponent. The general form of a logarithm is: log base b of a = c, which can be rewritten in exponential form as b^c = a. In simpler terms, the base (b) raised to the power of the result (c) gives us the argument (a) of the logarithm. This transformation is crucial because it allows us to get rid of the logarithm and work with a more familiar exponential equation. For example, if we have log base 2 of 8 = 3, this translates to 2^3 = 8. See how that works? We're just switching the equation into a form that's easier to manipulate.

Applying the Conversion to Our Problem

Now, let's apply this knowledge to our specific problem: log base 3 of (x+3) = 2. Using the conversion we just discussed, we can rewrite this equation in exponential form. The base is 3, the exponent (or result) is 2, and the argument is (x+3). So, following our formula (b^c = a), we get 3^2 = (x+3). This simple transformation is the magic key that unlocks the rest of the solution. We've taken a seemingly complex logarithmic equation and turned it into a straightforward algebraic equation. From here, it's just a matter of basic math to find the value of x. Isn't that neat how we can change the form of the equation to make it easier to solve? Keep this trick in your toolkit, as it’s super handy for tackling log problems!

Solving the Equation

Now that we've converted our logarithmic equation into exponential form, let's roll up our sleeves and solve for x! Remember, we have the equation 3^2 = (x+3). This is a much simpler equation to handle, and we're going to break it down step by step to make sure everyone's on the same page.

Simplifying the Exponential Term

The first thing we need to do is simplify the exponential term, which is 3^2. Guys, this simply means 3 multiplied by itself: 3 * 3. So, 3^2 equals 9. Replacing 3^2 with 9 in our equation, we now have 9 = (x+3). See how we're making progress? By simplifying each part of the equation, we're getting closer and closer to isolating x. This step is fundamental because it reduces the complexity of the equation, making it easier to see the next step. So, always start by simplifying any exponential terms you encounter.

Isolating the Variable

Our next goal is to isolate x, meaning we want to get x all by itself on one side of the equation. Currently, we have 9 = (x+3). To get x by itself, we need to get rid of the +3 on the right side. We can do this by performing the opposite operation, which is subtraction. So, we subtract 3 from both sides of the equation. This is super important: whatever you do to one side of the equation, you must do to the other side to keep it balanced. Subtracting 3 from both sides gives us: 9 - 3 = (x+3) - 3. Simplifying this, we get 6 = x. Hooray! We've successfully isolated x and found its value. This step demonstrates a fundamental principle in algebra: maintaining balance while manipulating equations.

The Solution

So, after simplifying the exponential term and isolating the variable, we've arrived at our solution: x = 6. That wasn't so bad, right? We took a logarithmic equation, converted it to exponential form, and then used basic algebraic principles to find the value of x. This is a typical process for solving logarithmic equations, so mastering these steps will help you tackle a wide range of problems. Always remember to convert, simplify, and isolate! These are your key ingredients for success in solving equations. Pat yourselves on the back, guys, you've just conquered a logarithmic equation!

Checking the Solution

Before we celebrate our victory, it's always a smart move to check our solution. This is a crucial step, guys, because it helps us ensure that our answer is correct and doesn't introduce any errors into the original equation. In the context of logarithmic equations, checking is particularly important because logarithms have domain restrictions. This means that certain values of x might make the logarithm undefined (like trying to take the logarithm of a negative number or zero). So, let's double-check our answer to make sure it's valid.

Substituting the Value of x

To check our solution, we'll substitute the value we found (x = 6) back into the original logarithmic equation: log base 3 of (x+3) = 2. Replacing x with 6, we get log base 3 of (6+3) = 2. Now, let's simplify the expression inside the logarithm. We have 6+3, which equals 9. So, our equation now looks like this: log base 3 of 9 = 2. We're getting closer to verifying our answer. Remember, what we're trying to see is if this statement is true. If it is, then our solution is correct!

Verifying the Logarithmic Expression

Now, we need to verify if log base 3 of 9 is indeed equal to 2. To do this, we can think back to the definition of a logarithm. We're asking, “To what power must we raise 3 to get 9?” Well, we know that 3 squared (3^2) is equal to 9. So, log base 3 of 9 is indeed 2. Our equation checks out! Guys, this confirms that our solution, x = 6, is correct. This process of checking might seem like an extra step, but it's an invaluable practice that can save you from making mistakes. Always take the time to verify your answers, especially in math problems!

Conclusion

Alright, guys, we've successfully solved the logarithmic equation log base 3 of (x+3) = 2! We walked through the steps of converting the equation to exponential form, simplifying, isolating the variable, and, most importantly, checking our solution. Remember, the key to mastering these types of problems is to understand the relationship between logarithms and exponents and to apply the rules of algebra consistently. So, next time you encounter a logarithmic equation, don't sweat it! Just follow these steps, and you'll be solving them like a pro in no time. Keep practicing, and you'll build your confidence and skills. You've got this! And remember, x = 6 is our final, verified answer. Great job, everyone!